2. Modal Principles of Jeﬀrey Updating

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c 2018 The Author(s)

https://doi.org/10.1007/s11787-018-0205-8 Logica Universalis

On the Modal Logic of Jeﬀrey Conditionalization

Zal´an Gyenis

In memory of Marcin Mostowski.

Abstract. We continue the investigations initiated in the recent papers (Brown et al. in The modal logic of Bayesian belief revision,2017; Gyenis in Standard Bayes logic is not finitely axiomatizable,2018) where Bayes logics have been introduced to study the general laws of Bayesian belief revision. In Bayesian belief revision a Bayesian agent revises (updates) his prior belief by conditionalizing the prior on some evidence using the Bayes rule. In this paper we take the more general Jeffrey formula as a conditioning device and study the corresponding modal logics that we call Jeffrey logics, focusing mainly on the countable case. The containment relations among these modal logics are determined and it is shown that the logic of Bayes and Jeffrey updating are very close. It is shown that the modal logic of belief revision determined by probabilities on a finite or countably infinite set of elementary propositions is not finitely axiomatizable. The significance of this result is that it clearly indicates that axiomatic approaches to belief revision might be severely limited.

Mathematics Subject Classiﬁcation. Primary 03B42, 03B45; Secondary 03A10.

Keywords.Modal logic, Bayesian inference, Bayes learning, Bayes logic, Jeﬀrey learning, Jeﬀrey conditionalization.

1. Background and Overview

This paper continues the investigations initiated in the recent papers [7,11]

where Bayes logics have been introduced to study the modal logical properties

Presented atUnilog’2018, Vichy, France, as the winner of theAlfred Tarski Logic Prize 2018and candidate for theUniversal Logic Prize 2018.

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of statistical inference (Bayesian belief revision) based on Bayes conditionalization.

Suppose (X,B, p) is a probability space where the probability measure pdescribes knowledge of statistical information of elements ofB. In the terminology of probabilistic belief revision one says that elements inBstand for the propositions that an agent regards as possible statements about the world, and the probability measureprepresents an agent’sprior degree of beliefs in the truth of these propositions. Belief revision is about to learn new pieces of information: Learning propositionA ∈ B to be true, the agent revises his priorpon the basis of this evidence and replacespwith some new probability measureq(often calledposterior) that can be regarded as the probability measure that the agentinfers from p on the basis of the information (evidence) that A is true. This transition from pto q is what is called statistical inference. We say in this situation that “qcan be learned fromp”¹ and that “it is possible to obtain/learn q from p”. This clearly is a modal talk and calls for a logical modeling in terms of concepts of modal logic. Indeed, the core idea of the paper [7] was to look statistical inference as an accessibility relation between probability measures: the probability measureqcan be accessed from the probability measurepif for some evidenceAwe can infer fromptoq. (For more motivation on how exactly modal logic come to the picture we refer to the introduction of [7]).

But how do we get q? One possible answer is a fundamental model of statistical inference, the standard Bayes model that relies on Bayes conditionalization of probabilities: given a prior probability measurepand an evidence A ∈ B the inferred measure q is deﬁned by conditionalizingp upon A using the Bayes rule:

q(H)=. p(H |A) =p(H∩A)

p(A) (∀H ∈ B) (1.1) providedp(A)= 0. Whenq can be obtained frompusing Bayes conditionalization upon some evidenceA we say that qis Bayes accessible from p. The paper [7] studied the logical aspects of this type of inference from the perspective of modal logic and also hints that a similar analysis could be carried out when Bayes accessibility is replaced by the more general accessibility based on Jeﬀrey conditionalization.

Indeed, Bayesian belief revision is just a particular type of belief revision:

Various rules replacing the Bayes rule have been considered in the context of belief change, and one important particular type is Jeﬀrey conditionalization (see [9,32]). Jeﬀrey conditioning is a way of inferring to a new probability q from the prior probabilitypand from anuncertain evidence ri assigned to a

1This terminology is common in the literature of machine learning or artiﬁcial intelligence [3, 22], and it might be slightly confusing because one also says the “Agent learns theevidence”.

But the conceptual structure of the situation is clear: The Agent’s “learning”qmeans the Agentinfersqfrom some evidence (using conditionalization as inference device, see later).

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ﬁnite²partition{Ei}i<nofX (ri≥0,

i<nri= 1,p(Ei)>0) by making use of the Jeﬀrey rule:

q(H)=.

i<n

p(H |Ei)ri (1.2)

Jeﬀrey conditioning provides a more general method than Bayesian conditioning: if we assume that an element of the partition becomes certain (i.e.

ri = 1 for some indexi), then the Jeffrey rule (1.2) reduces to the Bayes rule q(H) =p(H|Ei). On this basis the Bayes rule is a special case of the Jeffrey rule. Taking the Jeffrey rule as an inference device gives rise to what we call Jeffrey accessibility: we say thatq can be Jeffrey accessed frompif qcan be obtained from p using (1.2) with some uncertain evidence. The aim of the current paper is to study the modal logical character of Jeffrey accessibility in a similar manner as it had been done in [7,11] with Bayes accessibility.

For monographic works on Bayesianism we refer to [6,19,32]; for papers discussing basic aspects of Bayesianism, including conditionalization, see [14, 16–18,30,31]; for a discussion of Jeﬀrey’s conditionalization, see [10].

Two remarks are in order here. First, in the literature of probabilistic updating apart from the Bayes and Jeffrey rules various other rules have been studied to update a prior probability, such as entropy maximalization or minimalization principles among others. Conditionalizing is a concept and technique in probability theory that is much more general than the Bayes rule (1.1) (also called “ratio formula” [25]). Both the Bayes rule and Jeffrey rule are special cases of conditioning with respect to a σ-field, see [4, Chapters 33–34] and [13] for further discussion of the relation of Bayes and Jeffrey rules to the theory of conditionalization via conditional expectation determined by σ-fields. We refer to [9] for a comparison of such methods.

Second, let us note here that there is a huge literature on other types of belief revision as well. Without completeness we mention: the AGM postu- lates in the seminal work of Alchurrón–Gärdenfors–Makinson [1]; the dynamic epistemic logic [29]; van Benthem’s dynamic logic for belief revision [28]; probabilistic logics, e.g. Nilsson [23]; and probabilistic belief logics [2]. Typically, in this literature beliefs are modeled by sets of formulas defined by the syntax of a given logic and axioms about modalities are intended to prescribe how a belief represented by a formula should be modified when new information and evidence are provided. Viewed from the perspective of such theories of belief revision our intention in this paper, following [7], is very different. We do not try to give a plausible set of axioms in some nicely designed logic to capture desired features of (probabilistic) belief revision. On the contrary, we take the model that is actually used in applications of probabilistic learning theory and aim at an in-depth study of this model from a purely logical perspective. Bayesian probabilistic inference is relevant not only for belief change:

2Finiteness of the partition does not play a crucial role here, in fact, it turns out from Sect.

3that from the modal logical point of view allowing inﬁnite (countable) partitions does not make any diﬀerence. See the discussion in Sect.3.

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Bayes and Jeﬀrey conditionalization are the typical and widely applied inference rules also in situations where probability is interpreted not as subjective degree of belief but as representing objective matters of fact. Finding out the logical properties of such types of probabilistic inference has thus a wide in- terest going way beyond the conﬁnes of belief revision.

Below we recall the most important preliminary deﬁnitions from [7] and deﬁne the central subjects of the present paper. Concerning notions in modal logic we refer to the books Blackburn–Rijke–Venema [5] and Chagrov–

Zakharyaschev [8]. We take the standard unimodal language given by the grammar

a | ⊥ | ¬ϕ | ϕ∧ψ | ♦ϕ (1.3) deﬁning formulasϕ, whereabelongs to a nonempty countable set Φ of propositional letters. We use more-or-less standard notation and terminology but to be on the safe side the most basic concepts are recalled in the “Appendix”.

Formal background. For a measurable space X,B we denote by M(X,B) the set of all probability measures over X,B. M(X,B) serves as the set of

“possible worlds” in the Kripkean terminology and Bayes accessibility relation has been deﬁned in [7] as follows: Forv, w∈M(X,B) we say thatwisBayes accessiblefromvif there is anA∈ Bsuch thatw(·) =v(· |A). We denote the Bayes accessibility relation onM(X,B) byR(X,B). [7] introduces the notion of Bayes frames and Bayes logics:

Deﬁnition 1.1 (Bayes frames). A Bayes frame is a Kripke frameW, Rthat is isomorphic, as a directed graph, to F(X,B) = M(X,B), R(X,B) for a measurable spaceX,B.

For convenience, we rely on the convention that elementary events{x}for x∈X always belong to the algebraB; the reader can easily convince himself that this convention can be bypassed for the purposes of this paper. As a result, note that if the measurable spaceX,Bis ﬁnite or countably inﬁnite, thenB must be the powerset algebra℘(X). Therefore, in the countable case (i.e. when X is countable) instead of writing F(X, ℘(X)),M(X, ℘(X)) or R(X, ℘(X)) we sometimes simply writeF(X),M(X) orR(X), respectively.

Definition 1.2 (Bayes logics). A family of normal modal logics have been defined in [7] based on finite or countable or countably infinite or all Bayes frames as follows.

BL<ω={φ: (∀n∈N)F(n, ℘(n))φ} (1.4)

BLω={φ: F(ω, ℘(ω))φ} (1.5)

BL_≤ω=BL<ω∩BLω (1.6)

BL={φ: (∀Bayes frames F)Fφ} (1.7) We callBL<ω (resp.BL_≤ω) the logic of ﬁnite (resp. countable) Bayes frames;

however, observe that the set of possible worlds M(X,B) of a Bayes frame F(X,B) is ﬁnite if and only if X is a one-element set, otherwise it is at least of cardinality continuum.

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Bayes logics in Deﬁnition1.2capture the laws of Bayesian learning:BL<ω

is the set of general laws of Bayesian learning based on all ﬁnite Bayes frames, while thegeneral laws of Bayesian learningindependent of the particular rep- resentation X,B of the events is then the modal logic BL. The following theorem has been proved in [7].³

Theorem 1.3. The following (non)containments hold.

• S4⊆BL⊆BLω=BL≤ωBL<ω,

• S4.1BLω,

• S4.1+GrzBL_<ω.

The logic of ﬁnite Bayes frames has completely been described in [7] and, in particular, it has been shown that

• BL<ω has the ﬁnite frame property [7, Proposition 5.8],

• BL<ω isnotﬁnitely axiomatizable [7, Propositions 5.9].

In a similar manner we define Jeffrey accessibility: Given two measures p, q ∈ M(X,B) we say that q is Jeffrey accessible from pif there is a finite partition{Ei}i<nand uncertain evidenceri assigned to this partition (ri≥0,

i<nri = 1,p(Ei)>0) such that Eq. (1.2) holds. Denote the corresponding accessibility relation byJ(X,B).

Definition 1.4 (Jeffrey frames). A Jeffrey frame is a Kripke frameW, Rthat is isomorphic, as a directed graph, to J(X,B) = M(X,B), J(X,B) for a measurable spaceX,B.

A remark similar as above applies here: if the underlying set X of the measurable space X,B is countable, then we may write J(X) and J(X) instead of the longerJ(X, ℘(X)) andJ(X, ℘(X)).

Definition 1.5 (Jeffrey logics). A family of normal modal logics is defined for a cardinalκand∈ {=, <,≤}as follows.

JLκ={φ: (for allX,Bwith|X|κ) J(X,B)φ} (1.8) JL={φ: (∀Jeffrey frames J)J φ} (1.9) We callJL<ω (resp.JLω) the logic of finite (resp. countable) Jeffrey frames or sometimes we use the term “finite (resp. countable) Jeffrey logic”.

Jeffrey logics in Definition1.5capture the laws of Jeffrey updating:JL<ω

is the set of general laws of Jeffrey learning based on all finite Jeffrey frames, while thegeneral laws of Jeffrey learning independent of the particular repre- sentationX,Bof the events is then the modal logicJL.

From the point of view of applications of probabilistic updating the most important classes of Bayes and Jeffrey frames are the ones determined by measurable spacesX,Bhaving a finite or a countably infiniteX. Taking the first steps, this paper focuses only on the case with countableX, nevertheless, ques- tions similar to what we ask here could be raised in connection with standard

3 Some of the basic terminology of modal logic, such as what S4 is, is recalled in the

“Appendix”.

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Borel spaces, e.g. when B is the Borel (or Lebesgue)σ-algebra over the unit intervalX= [0,1]. It seems that countability ofX serves as a dividing line and continuous spaces require diﬀerent techniques than the ones employed here (cf.

[11] where Bayes updating over standard Borel spaces was investigated).

Overview of the paper. Firstly, in Sect.2we discuss the connections of Jeffrey logics to a list of modal axioms that are often considered in the literature. In particular, Proposition2.1shows thatJLS4butJLM(thusJLS4.1), JL_≤ω S4.1 and JL_κ Grz for any ∈ {=, <,≤} and κ > 1. Then, Theorem2.2clarifies the containments between the different Jeffrey logics:

S4 ⊆ JL ⊆ JL_ω=JL_≤ω ⊆ JL_<ω ⊆ JL_n+k ⊆ JL_n (1.10)

In Sect.3 we prove that the logic of Jeffrey updating (in the countable case) coincide with the logic of absolute continuity (see Theorem 3.7). The interesting part is when X is countably infinite: as a side result it turns out that from the modal logical point of view it does not matter whether or not we allow infinite partitions in the Jeffrey formula (1.2). In other words, the general laws that apply to Jeffrey learning are the same in both cases (and coincide with that of absolute continuity).

In Sect.4 we ask the question “how close Bayes and Jeﬀrey logics are?”

It turns out that ﬁniteness of X serves as a dividing line: there is a proper containment

JLn BLn and JL<ω BL<ω (1.11)

The case with countably infinite X, however, seems to show a completely different behavior. Theorem4.6disqualifies a large class of normal modal logics L that can possibly be put in between JLω L BLω. We also show thatJLωis indistinguishable fromBLωwithin a large class of modal formulas (Corollary4.7). It seems that the standard techniques fail to make a distinction betweenJLωandBLω and thus we conjecture that they are indeed the same.

This we articulated in Problem4.8: Are the logicsJLωandBLωthe same?

Finally, Sect.5deals with finite axiomatizability of the logicsJL<ω and JLω. Theorem5.8 states that the logic of finite Jeffrey framesJL<ω is not finitely axiomatizable, while Theorem5.16claims the same non finite axiomatizability result forJLω (moreover countable Jeffrey logics are not axiomatizable byanyset of formulas using finitely many variables). The situation is thus similar to that of Bayes logics (recall thatBL<ω is not finitely axiomatizable, see [7, Propositions 5.9]). Such no-go results have a philosophical significance:

they tell us that there is no finite set of formulas from which all general laws of Bayesian belief revision and Bayesian learning based on probability spaces with a countable set of propositions can be deduced. Bayesian learning and belief revision based on such simple probability spaces are among the most important instances of probabilistic updatings because they are widely used in applications. If the axiomatic approach to belief revision is not capable to characterize the logic of the simplest, paradigm form of belief revision, then this casts doubt on the general enterprize that aims at axiomatizations of belief revision systems. The cases with finite or countably infiniteXrequire different techniques, therefore this section is divided into two subsections, accordingly.

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2. Modal Principles of Jeﬀrey Updating

In this section we discuss the connections of Jeﬀrey logics to a list of modal axioms that are often considered in the literature: theT, 4,M andGrzaxioms (see “Appendix”).

We claim first that each Jeffrey frame is anS4-frame, that is, the accessibility relation of the frame is reflexive and transitive. Take any Jeffrey frame J(X,B) = W, R. As we mentioned earlier Bayes conditioning is a special case of Jeffrey conditioning. This immediately implies reflexivity ofR as for all probability measures w ∈W we have w(·) = w(· | X). As for transitivity, supposeu, v, w∈W with uRv andvRw. Taking into account the Jeffrey formula (1.2), we need two partitions{E_i}and{F_j}and uncertain evidences ri and sj assigned to these partitions such that v(H) =

iu(H |Ei)ri and w(H) =

jv(H |Fj)sj. Checking transitivity ofR requires some eﬀorts but only basic algebra is involved (such as reordering sums) and thus we skip the lengthy calculations and only hint that one should take the common reﬁne- ment{Ei∩Fj}i,jof the two partitions with suitable valuesti,jcalculated from the valuesri andsj. ConsequentlyJ(X,B)S4and thereforeS4⊆JL.

AnS4-frame is anS4.1-frame if it validates the axiomM that requires the existence of endpoints: the frame J(X,B) = W, R validates M if and only ifR has endpoints in the following sense:

∀w∃u(wRu ∧ ∀v(uRv→u=v)) (2.1) IfX is countable, then the Dirac measuresδ{x} forx∈X are endpoints: take any u∈ W and pick x∈ X such that u({x})= 0. Then δ{x} =u( · | {x}).

It follows thatS4.1⊆JL_≤ω. (We will see later on that this containment is proper asJL_≤ω is not ﬁnitely axiomatizable).

On the other hand, we claim thatM /∈JLand consequentlyS4.1⊆JL.

To this end it is enough to give an example for a Jeﬀrey frame J(X,B) in which there are paths without endpoints. Consider the frameJ([0,1],B) where [0,1] is the unit interval andBis the Borelσ-algebra. Then, for the Lebesgue measurewwe have

J([0,1],B)|=∃u(wRu ∧ ∀v(uRv→u=v)) (2.2) We note that none of the logics JLκ (for ∈ {=, <,≤} and κ >1) contain the Grzegorczyk axiom Grz as a Jeﬀrey frame J always contains a complete subgraph of cardinality continuum.

Summing up we get the following proposition.

Proposition 2.1. The following statements hold:

• JLS4 butJLM, in particular JLS4.1.

• JL≤ωS4.1.

• JLκGrz for any ∈ {=, <,≤} andκ >1.

The containments between different Jeffrey logics are clarified in the next theorem.

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Theorem 2.2. The following containments hold.

S4 ⊆ JL ⊆ JLω=JL_≤ω ⊆ JL<ω ⊆ JL_n+k ⊆ JLn (2.3)

Proof. From the very deﬁnition the following containments are straightforward:

JL ⊆ JL_≤ω ⊆ JL<ω ⊆ JLn and JL ⊆ JL_≤ω ⊆ JLω (2.4)

Next we show JLm ⊆ JLn for m > n and JLω ⊆ JL<ω. The proof relies on Lemma2.3. If X,Band Y,S are measurable spaces, then we say that X,B can be embedded intoY,S (X,B → Y,Sin symbols) if there is a surjective measurable function f : Y → X such that f⁻¹ : B → S is a σ-algebra homomorphism.

Lemma 2.3. IfX,B→ Y,S, then J(Y,S)J(X,B)

Proof. Letf : Y → X be a surjective measurable function (f⁻¹ : B → S is a σ-algebra homomorphism). For a probability measure p ∈ M(Y,S) let us assign the probability measureF(p)∈M(X,B) deﬁned by the equation

F(p)(A) =p

f⁻¹(A)

(A∈ B) (2.5)

ThenF :J(Y,S)J(X,B) is a surjective bounded morphism.

Now, form > nwe haveJ(m)J(n) and J(ω)J(n). Hence, the containmentsJLm⊆JLn form > nandJLω⊆JL<ω follow. We also obtain

JLω=JL_≤ω asJL_≤ω=JLω∩JL<ω.

3. Relation to Absolute Continuity

Considering Eq. (1.2) [or even Eq. (1.1)] it is easy to see thatqhas value 0 on every element H ∈ B which has p-probability zero. The technical expression of this is thatqis absolutely continuous with respect top. Therefore absolute continuity is necessary for Bayes or Jeﬀrey accessibility. In general, forp, q∈ M(X,B) we say that q is absolutely continuous with respect to p(q pin symbols) ifp(A) = 0 impliesq(A) = 0 for allA∈ B.

Let us now assume thatX={x0, . . . , xn−1}is ﬁnite and take any probability measurep∈M(X, ℘(X)). Ifq∈M(X, ℘(X)) is a probability measure such that q p, then by taking the partition Ei = {xi} for i < n and the uncertain evidenceri=q(Ei), we get

q(H) =

i<n

p(H |Ei)ri (3.1)

for allH ⊆ X. This means that given any prior probability p and an other probabilityqthat is absolutely continuous with respect top, if the probability space is finite, then q can be obtained from p by the Jeffrey rule. In other words, absolute continuity and Jeffrey accessibility coincide in the finite case.

This motivates us to introduce Kripke frames where the accessibility relation is deﬁned by absolute continuity, as follows.

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Deﬁnition 3.1. For a probability spaceX,Bwe deﬁne the Kripke frame A(X,B) =

M(X,B),

(3.2) wherestands for absolute continuity: For probability measuresp, q∈M(X,B) we writepq(orqp) ifp(A) = 0 impliesq(A) = 0 for allA∈ B.

Definition 3.2 (Logics of Absolute Continuity). In a similar manner to Defini- tions1.2and1.5we define a family of normal modal logics based on absolute continuity. Letκbe a cardinal and∈ {=, <,≤}.

ACLκ ={φ: (for allX,Bwith|X|κ) A(X,B)φ} (3.3)

ACL={φ: (∀X,B) A(X,B)φ} (3.4)

Our observation at the beginning of this section proves the next proposition.

Proposition 3.3. JLn=ACLn andJL<ω=ACL<ω for any n∈N.

Proof. For a ﬁniteX a probability measureq∈M(X, ℘(X)) can be obtained fromp∈M(X, ℘(X)) by means of Jeﬀrey conditionalizing if and only ifpq. This implies that the frames A(X) and J(X) are identical. Consequently ACLn= Λ(A(n)) = Λ(J(n)) =JLn, andACL<ω=

nACLn =

nJLn=

JL<ω.

What about the countably infinite case? The answer depends on whether or not we allow infinite partitions in the Jeffrey formula (1.2).

If we allow infinite partitions in the Jeffrey formula (1.2) andX is countably infinite, say X =N, then taking the partition Ei = {i} and the values ri =q({i}) fori∈N, Jeffrey formula leads to

q(H) =

i∈N

p(H |Ei)ri (3.5)

for all H ∈ ℘(N), provided p, q ∈ M(N, ℘(N)) are such that p q. This immediately ensures that q is Jeﬀrey accessible from p if and only if q is absolutely continuous with respect top, and in particularJLω=ACLω.

There are good reasons, however, to keep the partition in the Jeffrey formula finite. The requirement that the uncertain evidence is given by a probability measure on aproper,non-trivialpartition can be important: otherwise, as we have seen it, every probability measure can be obtained from itself as evidence—a triviality. The recent paper [12] argues that even in the finite case (i.e. whenX is finite) it makes sense not to consider the trivial partition in the Jeffrey rule [as we did in Eq. (3.1)]. However, by sticking to all proper partitions in Jeffrey accessibility we would lost transitivity⁴which is a well-desired property in the context of learning theory. The natural way to overcome this problem is to not allow all proper partitions but rather just the restricted set offinite partitions. This way we can keep transitivity and also, as we will shortly see, the infinite Jeffrey logic JLω will still coincide with ACLω. In other words, from the logical point of view whether or not we allow infinite

4The common reﬁnement of two proper partitions can lead to the trivial partition, see the example in Figure 4 in [12].

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partitions in the Jeﬀrey rule (1.2) does not make any diﬀerence. The rest of this section is devoted to prove this statement.

Recall that for a countableX, the support of a probability measureu∈ M(X, ℘(X)) is the set supp(u) ={x∈X : u({x})= 0}.

Lemma 3.4. Letp, q, rbe probability measures over the measure spaceN, ℘(N) and suppose that both q and r are Jeﬀrey accessible from p and supp(q) = supp(r). Thenr is Jeﬀrey accessible fromq and vice versa.

Proof. Letp, qandrbe as in the statement. According to Proposition7.2, as bothqandr are Jeﬀrey accessible fromp, we have that the Radon–Nikodym derivatives ^dq_dp and _dp^dr are step functions p-almost everywhere. As supp(q) = supp(r),q andrare mutually absolutely continuous. In order to get thatris Jeﬀrey accessible fromq, it is enough (by Proposition7.2again) to check that the Radon–Nikodym derivative ^dr_dqis a step function,q-almost everywhere. But

dr

dq =^dr_dp·^dp_dq except for aq-measure zero set, and it is straightforward that the product of two step functions is a step function. Thatq is Jeﬀrey accessible

fromris completely similar.

Proposition 3.5. JLω⊆ACLω.

Proof. It is enough to prove thatA(ω)J(ω). Indeed, we claim that whenever p∈ J(ω) is a faithful measure (meaning that it has full support supp(p) =ω), then the generated subframeJ^p is isomorphic toA(ω). For this we only need that ifq andrare Jeﬀrey accessible from pand supp(q) = supp(r), thenr is Jeﬀrey accessible fromqand vice versa. This exactly is Lemma3.4.

Proposition 3.6. JLω⊇ACLω.

Proof. It is enough to prove that A(ω)J(ω) for a suitable disjoint union A(ω). Indeed, recall that A(ω)J(ω) implies Λ

A(ω)

⊆Λ A(ω)

⊆ Λ

J(ω)

, that is,ACLω⊆JLω. The construction is as follows.

For a non-empty subset X ⊆ ω consider those probability measures in J(ω) whose support is X and write

SX =

u∈M(ω, ℘(ω)) : supp(u) =X

. (3.6)

As the Jeﬀrey accessibility relation is transitive, SX can be partitioned into clicks. A click is a maximal subset K of SX such that any two u, v ∈K are mutually Jeﬀrey accessible. Let Kα^X be an enumeration of the clicks of SX

(α < κX for some cardinalκX depending onX). Note that each K_α^X is either a 1-element set or has continuum many elements depending on whether or not X is a 1-element set.

For eachα < κω take a disjoint copyAα(ω) ofA(ω) and write A^X_α =

u∈ Aα(ω) : supp(u) =X

. (3.7)

Note that eachA^ω_α has continuum many elements.

Finally, take arbitrary bijections Fα : A^ωα → Kα^ω (for α < κω) and let F =

α<κωFα be the union of these bijections. As the copiesA^ωα are disjoint,

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F is a well-deﬁned bijection between

αA^ω_αandSω(this latter set is taken as a subset ofJ(ω)).

The content of Lemma3.4can be interpreted in our context as follows.

Take any probabilityp∈ J(ω) with supp(p) =X. SupposeY ⊆X is a non- empty subset andq, rare measures inJ(ω) such that supp(q) = supp(r) =Y and bothq and rcan be Jeﬀrey accessed from p. Thenq and rmust belong to the same click of SY. It follows that F can be extended from

αA^ω_α to the entire

αA(ω)_α in a homomorphic way. Checking that this extension is indeed a bounded morphism is not hard and left to the reader.

Summing up, independently of whether or not we allow inﬁnite partitions in the Jeﬀrey formula (1.2) we obtained the following theorem.

Theorem 3.7. For all countable cardinals κand∈ {=, <,≤}we have

JL_κ=ACL_κ. (3.8)

Proof. The equationsJLn=ACLn forn∈ω andJL<ω=ACL<ωis Propo- sition3.3. Combining Propositions3.5and 3.6we getJLω=ACLω. Finally, JL≤ω=ACL≤ω follows from the previous results and the deﬁnition.

This result enables us to use the frames A(n) and A(ω) instead of the more complex Jeﬀrey framesJ(n) andJ(ω).

4. How Close Bayes and Jeﬀrey Logics Are?

One of the main results in [7] is Theorem 5.2 which relates Bayes logics to the strongest modal companion of Medvedev’s logic of finite problems. We start by recalling definitions and theorems from [7,26]. Medvedev’s logic of finite problems and its extension to infinite problems by Skvortsov originate in intuitionistic logic. (For an overview we refer to the book [8] and to Shehtman [26]; Medvedev’s logic of finite problems is covered in the papers [15,20,21,24, 26,27].)

Deﬁnition 4.1 (Medvedev frames). A Medvedev frame is a frame that is isomorphic (as a directed graph) to℘(X){∅},⊇ for a non-emptyﬁnite set X.

For convenience, as a slight abuse of notation, we will call every frame of the form ℘(X){∅},⊇ (X being ﬁnite or inﬁnite) a Medvedev frame and we will use the notation

P_X⁰ =℘(X){∅},⊇ (4.1)

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A hierarchy or normal modal logics that correspond to the framesP_X⁰ can be given:

MLn=

φ: P_n⁰ φ

(4.2) ML<ω=

φ: (∀n∈N)P_n⁰φ

(4.3) MLω=

φ: P_ω⁰ φ

(4.4)

ML_≤ω=ML_<ω∩ML_ω (4.5)

ML=

α

MLα (4.6)

Observe that for α < β we have P_α⁰P_β⁰, consequently MLβ ⊆MLα. Since there are countably many modal formulas and proper class many cardinals, there must exists a cardinalα0 such that the sequence MLα stabilizes, i.e.ML=MLα0 or equivalently for allβ ≥α0 we haveMLβ=MLα0. Theorem 5.2 of [7] states the containments below.

ML⊆MLω =ML≤ωML<ωMLn

= = = =

BL BLω =BL≤ω BL<ω BLn

(4.7) A consequence of this result is that when the underlying set X of the measurable space is countable we can use the more easy-to-handle Medvedev frames instead of Bayes frames.

Lemma 4.2. For a countableX the mappingf :A(X)P⁰(X)deﬁned by

f(p) = supp(p) (4.8)

is a surjective bounded morphism.

Proof. Surjectivity of f is straightforward. f is a homomorphism (preserves accessibility) because for p, q ∈ M(X, ℘(X)) we have p q if and only if supp(p)⊇supp(q). To verify the zig-zag property, suppose supp(p)⊇A. We needq∈M(X, ℘(X)) such thatpq and supp(q) =A. Finding such aq is easy, take for example the conditional probabilityq(·) =p(· |A).

Corollary 4.3. ACLκ ⊆ MLκ holds for∈ {=, <,≤} andκcountable.

Proof. Immediate from Lemma4.2.

Corollary 4.4. JL_κ ⊆ BL_κ holds for∈ {=, <,≤} andκcountable.

Proof. Combine Corollary4.3, Theorem 3.7and Theorem 5.2. in [7].

We note that none of the logicsACLn (forn >1) contain the Grzegor- czyk axiomGrz as A(n) always contain a complete subgraph of cardinality continuum. It is easy to check that Medvedev frame over a ﬁnite set P⁰(n) validateGrz, that is,Grz∈BL<ω. Therefore we get

To have all the containments between Bayes and Jeﬀrey logics the only question remained is whetherJLω = BLω. (By Corollary4.4 we know that JLω⊆BLω.) The Grzegorczyk axiom does not diﬀerentiate betweenJLωand

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BLω as none of these logics contain the formulaGrz. In fact, we prove that JLω is indistinguishable from BLω within a large class of modal formulas.

On the other hand the standard techniques (generated subframes, bounded morphisms) to prove the equality of the two logics do not seem to work: none ofA(ω)P⁰(ω) or P⁰(ω)A(ω) holds.

We call a Kripke frame F click free if there are no two diﬀerent worlds in F that are mutually accessible, i.e. the largest click in F has size at most one. Note that click freeness enables reﬂexive points.

Lemma 4.5. Let F be a click free S4-frame. Then P⁰(ω)F if and only if A(ω)F.

Proof. (⇒) The claim thatP⁰(ω)F impliesA(ω)F is straightforward as any bounded morphism f : P⁰(ω) F can be lifted up to a bounded morphismf⁺:A(ω)F by lettingf⁺(p)=. f(supp(p)) forp∈M(ω, ℘(ω)).

(⇐) That A(ω) F impliesP⁰(ω) F can be veriﬁed by observing that click freeness of F ensures that all points of a click in A(ω) must be mapped to the same point ofF. Thus any morphismf :A(ω) F can be pushed down to a morphism f⁻ : P⁰ F by letting for all ∅ = X ⊆ ω, f⁻(X)=. f(p) for anyp∈M(ω, ℘(ω)) with supp(p) =X. Theorem 4.6. There is no normal modal logic L such that

JLω L BLω (4.10)

andLis the logic of a click free S4-frame F with A(ω)F.

Proof. Immediate from Lemma4.5.

The previous theorem tells us that if we would like to distinguish JLω

fromBLω, then the standard technique of ﬁnding a bounded morphic image of A(ω) that does the distinction fails (provided that this bounded morphic image is transitive and click free). We note that every modal formula is validated on a suitable ﬁnite, transitive, click free frame, thus Theorem 4.6 gives the impression that the two logics JLω and BLω coincide. Applying the same technique the next Corollary tells us thatJLω is indistinguishable from BLω

within a large class of modal formulas called Jankov—de Jongh formulas (cf.

Theorem7.1in the “Appendix”).

Corollary 4.7. For a transitive, click free frameF we have

χF∈JLω ⇔ χF ∈BLω (4.11)

Proof. Combine Lemma4.5with Theorem7.1.

We end this section by an open problem.

Problem 4.8. Are the logics JLω andBLω the same?

Part of the question in Problem 4.8 is this: Is there any frame F such thatA(ω)F butP⁰(ω)F? Such a frameF must contain a proper click (must not be click free).

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5. Non Finite Axiomatizability

For a natural numberl a logicLisl-axiomatizable if it has an axiomatization using only formulas whose propositional variables are amongp1, . . . , pl. Every ﬁnitely axiomatizable logic isl-axiomatizable for a suitablel: take lto be the maximal number of variables the ﬁnitely many axioms in question use.

The main message of this section is that countable Jeffrey logicsJL<ω, JLω andJL_≤ω are not finitely axiomatizable. In fact, it turns out from the proof that they cannot even be axiomatized with (possibly infinitely many) formulas using the same finitely many propositional letters. Thus, these logics are not finite schema axiomatizable either. The finite and the countably infinite cases require slightly different techniques, therefore we split the proof into two subsections, accordingly.

Also recall that the logic of countable Jeffrey frames is proved to be equal to that of absolute continuity (see Theorem 3.7). This allows us to use the framesA(X) of absolute continuity rather than the more complicated Jeffrey framesJ(X). Phrasing it differently: we in fact show that the logicsACL<ω, ACLω and ACL≤ω are not finitely axiomatizable and then refer to the fact that JLκ = ACLκ for all countable cardinals κ and ∈ {=, <,≤} (see Theorem3.7).

5.1. The Finite Case

We aim at proving ACL<ω is not finitely axiomatizable. We show first that ACL<ω is a logic offinite frames, thus it has the finite frame property.

For each k, n ∈ N we define the finite frame Ak(n) as follows. Take the frame A(n). For each non-singleton set A ⊆ n the frame A(n) contains a complete subgraph of cardinality continuum (measurespwith support supp(p) =A). Replace this infinite complete graph with the complete graph onk vertices and keep everything else fixed. A more precise definition is the following.

Deﬁnition 5.1. Let n, k > 0 be natural numbers. For each non-singleton set a∈℘(n)− {∅} take new distinct points [a]₁, . . . ,[a]_k, and for each singleton a∈℘(n) take [a]₁ =· · · = [a]_k to be a single new point. The set of possible worlds of the frameAk(n) is the set

Ak(n) =

[a]₁, . . . ,[a]_k: a∈℘(n)− {∅}

(5.1) For two points [a]_i,[b]_j ∈Ak(n) we deﬁne the accessibility relation→as

[a]_i→[b]_j if and only if a⊇b (5.2) Lemma 5.2. For allnandk we haveA(n)Ak(n).

Proof. For a measurep ∈ M(n) the support supp(p) is a non-empty subset of n, therefore [supp(p)]₁, . . ., [supp(p)]_k are elements of Ak(n). Take any mappingf :M(n)→Ak(n) such that

f(p) = [supp(p)]_i for somei∈ {1, . . . , k} (5.3)

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andf is a surjection. Such a mapping clearly exists as for eacha∈℘(n)− {∅}

we have

|{p: supp(p) =a}|= 2^ℵ⁰ > k (5.4) We claim thatf is a surjective bounded morphism:

Homomorphism. Take p, q ∈ M(n) and suppose f(p) = [supp(p)]_i, f(q) = [supp(q)]_j. Then p q if and only if supp(p) ⊇ supp(q) if and only if [supp(p)]_i→[supp(q)]_j.

Zag property. Assume f(p) → [a]_i for some a ∈ ℘(n)− {∅}. This can be the case if and only if supp(p)⊇a. By surjectivity of f there is q such that f(q) = [a]_i, whence supp(p)⊇supp(q) which meanspq. Lemma 5.3. For each modal formula ϕ there is k ∈ N such that A(n) ϕ impliesAk(n)ϕ.

Proof. We prove that ifϕ uses the propositional lettersp1, . . . , pk only, then A(n)ϕimpliesA₂^k(n)ϕ. IfA(n)ϕ, then there is an evaluationV such that the model A(n), V ϕ. The truth of a formula in a model depends only on the evaluation of the propositional letters the formula uses, therefore we may assume thatV is restricted top1,. . .,pk.

Forx∈ A(n) we deﬁne a 0–1 sequence of length kaccording to whether x∈V(pi) holds for 1≤i≤k:

Px(i) =

1 ifx∈V(pi)

0 otherwise. (1≤i≤k) (5.5)

As there are 2^k diﬀerent 0–1 sequences of length k, the number of possible Px’s is at most 2^k.

Take anysurjectivemapping f :A(n)→ A₂^k(n) such that

f(x) = [supp(x)]_i for somei∈ {1, . . . , k} (5.6) and forx, y∈ A(n) with supp(x) = supp(y) we have

Px=Py implies f(x) =f(y) (5.7) Such a mappingf must exist as for each non-singletona∈℘(n)− {∅}we have 2^k elements [a]₁, . . ., [a]₂k in A₂^k(n), and this is the number of the possible Px’s. Let us now deﬁne the evaluationV overA₂^k(n) by

V(pi) ={f(x) : x∈V(pi)} (5.8) for 1≤i≤k. Condition (5.7) ensures that ifxandyagree onp1, . . . , pk, then so do the imagesf(x) andf(y). Thus,V is well-deﬁned. Following the proof of5.2one obtains that

f :A(n), V A₂^k(n), V (5.9) is a surjective bounded morphism. AsA(n), V¬ϕwe arrive atA₂^k(n), V

¬ϕ. This meansA₂^k(n)ϕ.

Proposition 5.4. ACL<ω = _∞

n=1

_∞

k=1Λ (Ak(n)).

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Proof. By combining Lemmas5.2and5.3the equality ∞

n=1

Λ (A(n)) = ∞ n=1

∞ k=1

Λ (Ak(n)) (5.10) follows immediately. The left-hand side of the equation is the deﬁnition of

ACL<ω.

Corollary 5.5. Finite Jeﬀrey logicJL<ω has the ﬁnite frame property.

Proof. JL<ω=ACL<ω by Theorem3.7and thus Proposition5.4implies JL<ω =

∞ n=1

∞ k=1

Λ (Ak(n)) (5.11)

As each frameAk(n) is ﬁnite, the proof is complete.

Proposition 5.6. Let K be a class of ﬁnite, transitive frames, closed under point-generated subframes. For every ﬁnite, transitive, point-generated frame F we have

FΛ(K) if and only if ∃(G ∈K)GF. (5.12) Proof. (⇐) If there isG ∈K such thatGF, then Λ(K)⊆Λ(G)⊆Λ(F).

(⇒) By way of contradiction suppose G F for all G ∈ K. Then by Theorem7.1 we haveG χ(F) for all G ∈K, in particular,χ(F)∈Λ(K). It is straightforward to see thatFχ(F), thusF Λ(K).

Theorem 5.7. ACL<ω is not ﬁnitely axiomatizable.

Proof. A logic L is not ﬁnitely axiomatizable if and only if for any formula φ∈Lthere is a frameFφ such thatFφL butFφφ.

We will use the proof that the modal counterpart of Medvedev’s logic of finite problems, ML<ω, is not finitely axiomatizable. We refer to [26] where it has been proved that for each modal formula φ∈ML<ω there is a finite, transitive, point-generated frameGφ such thatGφφwhileGφML<ω. The construction therein is such thatGφ is click free.

We intend to show thatGφACL<ω. This is enough becauseACL<ω⊂ ML<ω. By Proposition 5.4 ACL<ω is the logic of the class K = {Ak(n) : n, k∈N}of ﬁnite, transitive frames, closed under point-generated subframes.

Therefore, to show Gφ ACL<ω, by Proposition 5.6 it is enough to prove that Gφ is not a bounded morphic image of any Ak(n). Suppose, seeking a contradiction, that there exists a bounded morphismf :Ak(n) Gφ. Then for eacha∈℘(n)− {∅}the elements [a]₁,. . ., [a]_k should be mapped into the same pointxainGφ. This is because the points [a]_iare all accessible from each other, while inGφthere are no non-singleton sets in which points are mutually accessible. It follows thatf induces a bounded morphismf^∗:P⁰(n)→ Gφfrom the Medvedev frameP⁰(n) intoGφ by letting f^∗(a) =xa fora∈℘(n)− {∅}.

But this is impossible asGφML<ω.

Theorem 5.8. Finite Jeﬀrey logic JL<ω is not ﬁnitely axiomatizable.

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Proof. JL<ω = ACL<ω by Theorem 3.7 and this latter logic is not ﬁnitely

axiomatizable by Theorem5.7.

5.2. The Countably Inﬁnite Case

To gain non ﬁnite axiomatizability results for the countable Jeﬀrey logicJLω

we follow the method presented in Shehtman [26] and we recall the most important lemmas that we make use of.

Deﬁnition 5.9 [26]. For m >0 andk >2 the Chinese lantern is the S4-frame C(k, m) formed by the set

{(i, j) : (1≤i≤k−2,0≤j≤1) or (i=k−1,1≤j≤m) OR (i=k, j= 0)} (5.13) with the accessibility relation being an ordering:

(i, j)≤(i, j) iﬀ (i, j) = (i, j) ORi > i (5.14) C(m, k) is illustrated on page 373 in [26], however, we will not need any particular information aboutC apart from two lemmas that we recall below.

Lemma 5.10 (Lemma 22 in [26]). Let φ be a modal formula using l variables and letm >2^l. ThenC(k, m)φimplies C(k,2^l)φ.

Lemma 5.11 (Lemma 24 in [26]). For anyn >1 we have C(2ⁿ,2ⁿ)ML<ω. Let F = W,≤ be a ﬁnite ordering (partially ordered set) and pick x∈W. y is an immediate successor ofxifx < y and there is no x < z < y. (As usual<means≤ ∩ =). The branch indexbF(x) is the cardinality of the set of immediate successors ofx, and the depthdF(x) is the least upper bound of cardinalities of chains inFwhose least element isx. Thus,dF(x) = 1 means thatxhas no immediate successors.F isduplicate-freeif it is ﬁnite, generated andbF(u)= 1 for anyu∈W (cf. Shehtman [26]).

Lemma 5.12. P⁰(ω)C(k,2^k).

Proof. This is essentially Lemma 17 in [26]. Note that C(k, m) is duplicate- free. The point u= (k,0) inC(k,2^k) has depth d(u) = k and branch index b(u) = 2^k. If there wereP⁰(ω)C(k,2^k), then by Lemma 17 in [26] we would

haveb(u)<2^d(u) which is impossible.

Theorem 5.13. LetLbe a normal modal logic with S4⊆L⊆ML<ω. Suppose that for everyl≥1 andk > l there isn≥ksuch that χ(C(k,2ⁿ))∈L. Then Lis not l-axiomatizable for any numberl.

Proof. By way of contradiction suppose L is l-axiomatizable, that is, L = S4+ Σ where Σ is a set of formulas that can use only the ﬁrstl propositional variables. Let k = 2^l. By assumption there is n ≥ k so that χ(C(k,2ⁿ)) ∈ L. That Σ axiomatizes L means that every formula inL can be derived (in the normal modal calculus) from a ﬁnite set of axioms from Σ. Therefore there is anl-formulaφ∈L such thatχ(C(k,2ⁿ))∈S4+φ. This implies, by Theorem 7.1(B), that C(k,2ⁿ)φ. Asn ≥k= 2^l> l, Lemma 5.10ensures C(k,2^l)φ. In particular,C(k,2^l)L.

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On the other hand Lemma5.10implies (ask= 2^l) thatC(k,2^l)ML<ω. By assumptionL ⊆ML<ω so it follows thatC(k,2^l)L which is a contra-

diction.

Corollary 5.14. Let F be a frame, L= Λ(F) and assume S4⊆L ⊆ML<ω. Suppose for any k ≥ 1 there is n ≥ k such that for all u ∈ F we have F^uC(k,2ⁿ). ThenLis not l-axiomatizable for any ﬁnite numberl. Proof. Under the given assumptions Theorem7.1(A) implies that for allk≥1 there isn≥kso thatχ(C(k,2ⁿ))∈L. Then Theorem 5.13applies.

Theorem 5.15. ACLωis not ﬁnitely axiomatizable (in fact, it is notl-axioma- tizable for any ﬁnite numberl).

Proof. We intend to apply Corollary5.14.ACLω= Λ(A(ω)) and the containments S4 ⊆ACLω ⊆ ML<ω hold (see Theorem 2.2). Write A =A(ω). In order to use Corollary 5.14 we only need to verify that for any k ≥ 1 there is n ≥ k such that for all u ∈ A we have Aû C(k,2ⁿ). It is easy to see that for all u ∈ A, Aû is isomorphic either to A(ω) or to A(n) depending on whether or notu has an infinite support. Therefore it is enough to check A C(k,2^k). AsC(k,2^k) is a transitive, click free frame, according to Lemma 4.5 if A C(k,2^k), then we also have P⁰(ω) C(k,2^k). But this latter is

impossible by Lemma5.12.

Theorem 5.16. Countable Jeffrey logic JLω is not finitely axiomatizable (in fact, it is notl-axiomatizable for any finite numberl).

Proof. JLω =ACLω by Theorem3.7 and this latter logic is not ﬁnitely ax-

iomatizable by Theorem5.15.

6. Closing Words and Further Research Directions

Our aim was to study the modal logical character of Jeﬀrey accessibility in a similar manner as it has been done in [7,11] concerning Bayes accessibility.

We have seen that the modal logic of Jeffrey learning always extendsS4, and extendsS4.1only if the underlying measurable space is countable (see Propo- sition2.1). Containments between the different Jeffrey logics were clarified in Theorem2.2:

S4 ⊆ JL ⊆ JLω=JL_≤ω ⊆ JL<ω ⊆ JL_n+k ⊆ JLn. (6.1) and the relations of Jeﬀrey logics to Bayes logics were also drawn in Sect.4:

Equality betweenJLωandBLωremained open. Theorem4.6and Corollary4.7 hints that they might be equal and in Problem4.8we ask whether the logics JLω andBLω coincide.

We regard Sect.5the main result of the paper. Theorem5.8states that the logic of finite Jeffrey framesJL<ωis not finitely axiomatizable, while Theo- rem5.16claims the same non finite axiomatizability result forJLω(moreover

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countable Jeffrey logics are not axiomatizable by any set of formulas using finitely many variables). The picture is thus analogous to that of Bayes logics, see [7, Propositions 5.9]. The significance of these results is that they clearly indicate that axiomatic approaches to belief revision might be severely limited.

It is a longstanding open question whether the strongest modal companion of Medvedev’s logic of ﬁnite problemsML<ω(and thus Bayes logicBL<ω) is recursively axiomatizable (see [8, Chapter 2]). Since the class of Medvedev frames is a recursive class of ﬁnite frames,BL_<ω is co-recursively enumerable.

It follows that ifML<ωis recursively axiomatizable, thenBL<ωis decidable.

According to Corollary5.5finite Jeffrey logicJL<ω has the finite frame property. The proof reveals thatJL<ωis a logic of a recursive class of finite frames, thusJL<ω is co-recursively enumerable, as well. We are not aware any similar result for JLω, neither do we know whether Jeffrey logics are recursively axiomatizable. We conjecture that recursive axiomatizability of Jeffrey logics would solve the similar question for Medvedev’s logic, thus the problem might be severely hard.

Problem 6.1. Are any of the logics JL<ω andJLω recursively axiomatizable?

Finally, we have already mentioned that in the literature of probabilistic updating apart from the Bayes and Jeﬀrey rules various other rules have been studied to update a prior probability, such as entropy maximalization or minimalization principles, among others. We do believe that a similar analysis should be carried out when Bayes or Jeﬀrey accessibility is replaced by some other accessibility relation based on these various probability updating principles.

Acknowledgements

The author is grateful to Miklós Rédei for all the pleasant conversations about this topic (and often about more important other topics). The author would like to acknowledge the Premium Postdoctoral Grant of the Hungarian Acad- emy of Sciences hosted by the Logic Department at Eötvös Loránd University, and the Hungarian Scientific Research Found (OTKA), Contract No. K115593.

Open Access. This article is distributed under the terms of the Creative Com- mons Attribution 4.0 International License (http://creativecommons.org/licenses/

by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

7. Appendix

ω is the least inﬁnite cardinal (that is, the set of natural numbers). By a frame we always understand a Kripke frame, that is, a structure of the form F=W, R, whereW is a non-empty set (of possible worlds) andR⊆W×W